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the power rule is used to differentiate the algebraic expressions of the form x^n.

first remark that the polynomial x^m+k has simple roots.

Key Overview & Analysis

Let us call the roots c_j for 1\leq j\leq m.

Therefore x^n/(x^m+k) rewrites as a sum of simple elements.

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what is the derivative of xn?

For the function f (x) = xn, n should not equal 0, for reasons which will become clear.

Product of exponentials with same base.

If we take the product of two exponentials with the same base, we simply add the exponents:

Xaxb = xa + b.

using the ratio test, $\dfrac{a_{n+1}}{a_n} = \dfrac{(n+1)x^{n+1}}{nx^n} = \dfrac{(n+1)x}{n}$.

So, the series converges when $|x|

$$f(x) = \sum_{n=1}^\infty n.